Logic and sets · Deductive reasoning · Information and knowledge · Perfect squares
Ales, Brane and Cvetka made a large pile of cards, writing on each card one of the numbers \(2, 3, 4, 5, 6, 7, 8\); every one of these numbers appears on many cards. Maja, who arrived later, picked three cards and stuck one on each of their foreheads. None of the three could see the card on his or her own forehead, only the two cards on the others. Maja then told them:
"The numbers on your foreheads are not all different, and the product of all three numbers is a perfect square."
Each of the three then tried, on their own, to work out the number on their own forehead. Could all three of them succeed?
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2005, 1. letnik, category A, problem 4. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source